Some problems • Given a pattern P = P[1..m], find all occurrences of P in a text S = S[1..n] Suffix Trees and • Another problem: – Given two strings S [1..n ] and S [1..n ] find their Suffix Arrays 1 1 2 2 longest common substring. • find i, j, k such that S 1[i .. i+k-1] = S2[j .. j+k-1] and k is as large as possible. • Any solutions? How do you solve these problems (efficiently)? Exact string matching Applications in Bioinformatics • Finding the pattern P[1..m] in S[1..n] can be • Multiple genome alignment solved simply with a scan of the string S in – Michael Hohl et al. 2002 O(m+n) time. However, when S is very long – Longest common substring problem and we want to perform many queries, it – Common substrings of more than two strings would be desirable to have a search • Selection of signature oligonucleotides for algorithm that could take O(m) time. microarrays • To do that we have to preprocess S. The – Kaderali and Schliep, 2002 preprocessing step is especially useful in • Identification of sequence repeats scenarios where the text is relatively constant – Kurtz and Schleiermacher, 1999 over time (e.g., a genome), and when search is needed for many different patterns. Suffix trees History of suffix trees • Any string of length m can be degenerated • Weiner, 1973: suffix trees introduced, linear- into m suffixes. time construction algorithm – abcdefgh (length: 8) • McCreight, 1976: reduced space-complexity – 8 suffixes: • h, gh, fgh, efgh, defgh, cdefgh, bcefgh, abcdefgh • Ukkonen, 1995: new algorithm, easier to describe • The suffixes can be stored in a suffix-tree and this tree can be generated in O(n) time • In this course, we will only cover a naive (quadratic-time) construction. • A string pattern of length m can be searched in this suffix tree in O(m) time. – Whereas, a regular sequential search would take O(n) time. 1 Definition of a suffix tree An example suffix tree • Let S=S[1..n] be a string of length n over a • The suffix tree for string: 1 2 3 4 5 6 fixed alphabet S. A suffix tree for S is a tree x a b x a c with n leaves (representing n suffixes) and the following properties: – Every internal node other than the root has at least 2 children – Every edge is labeled with a nonempty substring of S. – The edges leaving a given node have labels starting with Does a suffix tree different letters. always exist? – The concatenation of the labels of the path from the root to leaf i spells out the i-th suffix S[i..n] of S. We denote S[i..n] by Si. What about the tree for xabxa? Problem • The suffix tree for string: 1 2 3 4 5 • Note that if a suffix is a prefix of another suffix x a b x a we cannot have a tree with the properties defined in the previous slides. – e.g. xabxa The fourth suffix xa or the fifth suffix a won’t be represented by a leaf node. xa an a are not leaf nodes. Solution: the terminal character $ The suffix tree for xabxa$ • Note that if a suffix is a prefix of another suffix we cannot have a tree with the properties defined in the previous slides. – e.g. xabxa The fourth suffix xa or the fifth suffix a won’t be represented by a leaf node. • Solution: insert a special terminal character at the end such as $. Therefore xa$ will not be a prefix of the suffix xabxa. 2 Suffix tree construction Suffix tree construction - 2 • Start with a root and a leaf numbered 1, connected • If the mismatch occurs at the middle of an by an edge labeled S$. edge e = S[u ... v] • Enter suffixes S[2..n]$; S[3...n]$; ... ; S[n]$ into the – let the label of that edge be a ...a tree as follows: 1 l – If the mismatch occurred at character ak, then • To insert Ki = S[i..n]$, follow the path from the root create a new node w, and replace e by two edges matching characters of Ki until the first mismatch at S[u ... u+k-1] and S[u+k ... v] labeled by a ...a character K [ j ] (which is bound to happen) 1 k and i ak+1...al (a) If the matching cannot continue from a node, denote • Finally, in both cases (a) and (b), create a that node by w (b) Otherwise the mismatch occurs at the middle of an new leaf numbered i, and connect w to it by edge, which has to be split an edge labeled with Ki[j ... |Ki|] Example construction Example contd... • Let’s construct a suffix tree for xabxac$ • Inserting the fourth suffix xac$ will cause the first edge to be split: $ $ • Start with: $ • After inserting the second and third suffix: $ $ $ $ $ • Same thing happens for the second edge when ac$ is inserted. Example contd... Complexity of the naive construction • After inserting the remaining suffixes the tree • We need O(n-i+1) time for the ith suffix. will be completed: Therefore the total running time is: n åO(i) = O(n 2 ) 1 • What about space complexity? – Can also take O(n2) because we may need to store every suffix in the tree separately, – e.g., abcdefghijklmn 3 Storing the edge labels efficiently Suffix tree applet • Note that, we do not store the actual • http://pauillac.inria.fr/~quercia/documents- substrings S[i ... j] of S in the edges, but only info/Luminy - their start and end indices (i, j). 98/albert/JAVA+html/SuffixTreeGrow.html • Nevertheless we keep thinking of the edge labels as substrings of S. • This will reduce the space complexity to O(n) Using suffix trees for pattern matching Illustration • Given S and P. How do we find all occurrences of P in S? • T = xabxac • Observation. Each occurrence has to be a prefix of some – suffixes ={xabxac, abxac, bxac, xac, ac, c} suffix. Each such prefix corresponds to a path starting at the • Pattern P1: xa root. • Pattern P2: xb 1. Of course, as a first step, we construct the suffix tree for S. Using the naive method this takes quadratic time, but linear-time algorithms b c a x (e.g., Ukkonen’s algorithm) exist. x 6 a 2. Try to match P on a path, starting from the root. Three cases: a c b (a) The pattern does not match ? P does not occur in T c 5 x c b a x (b) The match ends in a node u of the tree. Set x = u. a c 4 c (c) The match ends inside an edge (v,w) of the tree. Set x = w. 3 2 3. All leaves below x represent occurrences of P. 1 Running Time Analysis Scalability • Search time: • For very large problems a linear time and – O(m+k) where k is the number of occurrences of space bound is not good enough. This lead to P in T and m is the length of P the development of structures such as Suffix – O(m) to find match point if it exists Arrays to conserve memory . – O(k) to find all leaves below match point 4 Effects of alphabet size on Two implementation issues suffix trees • Alphabet size • Generalizing to multiple strings • We have generally been assuming that the trees are built in such a way that – from any node, we can find an edge in constant time for any specific character in S • an array of size |S| at each node • This takes Q(m|S|) space. More compact representation Generalized suffix trees • We may try to be more compact taking only O(m) • Build a suffix tree for a set of strings S = {S1, …, Sz} space. • Some issues – At each node, have pointers to only the edges that are needed • Nodes in tree may corresponds to substrings of • This slows down the search time potentially multiple strings Si • How much? – compact edge labels: need 3 fields (start position, stop position, string) – typically the minimum of O(log m) or O(log |S|) with a binary tree representation. – leaf labels now a set of pairs indicating starting position • This effects both suffix tree construction time and and string later searching time against the suffix tree. Longest common substring problem Selecting probes for microarrays • Build a generalized suffix tree for S1$1S2$2. • Wikipedia: Oligonucleotides are short Here $1 and $2 are different new symbols not sequences of nucleotides (RNA or DNA), occurring in S1 and S2. typically with twenty or fewer base pairs. • Given a set of genomic sequences, the problem is to identify at least one • Mark every internal node of the tree with {1}, signature oligonucleotide (probe) for each sequence. These probes must {2}, or {1,2} depending on whether its path hybridize to only the desired sequence. The algorithm produces a GST from the reverse compliment of all the genomic sequences (candidate label is a substring of S1 and/or S2. probe sequences). Using the GST, the algorithm identifies all common substrings and rejects these regions because probes designed in them • Find the internal node which is labeled by would not be specific to a single genomic sequence. Criteria such as {1,2} and has the largest “string depth”. probe length are used to further prune this tree. • Example: (with the applet) • http://www.zaik.uni- – pessimist%mississippi$ koeln.de/bioinformatik/arraydesign.html.en 5 Suffix arrays Suffix arrays • Suffix arrays were introduced by Manber and Example of a suffix array for acaaacatat$ Myers in 1993 3 • More space efficient than suffix trees 4 • A suffix array for a string x of length m is an 1 array of size m that specifies the lexicographic 5 ordering of the suffixes of x.